Each stage has multiple missions. Each mission has an objective. Complete the objectives to clear the stage. Missions must be raced from top to bottom.
When you complete a mission you will receive a mission rank. Once all missions are complete an average will be calculated for your 'Best Rank'.
The first time you achieve an 'S' rank for the entire stage you'll receive bonus trophies.
World Rankings - click on the world ranking button on each stage to compare your score against the rest of the world.
Your overall stage 'Best Rank' score will be used toward your world ranking.